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Mean field theories of quantum spin
glasses
Antoine Georges
Olivier Parcollet
Nick Read
Subir Sachdev
Jinwu Ye
Talk online:
Sachdev
Classical Sherrington-Kirkpatrick model
H   J ij Si  S j
i j
Jij : a Gaussian random variable with zero mean
Si : a unit length n component vector
Two routes to quantization
A. Quantum rotor model
 1

 dSi 
Action =  d   
 H
 2 g i  d 

2
n=1: Ising model is a transverse field g
Spectrum at Jij=0

 j
1
 
j
2
 j
1
 
j
2
g

n=3: randomly coupled spin dimers
j
j



1
 ,
  
j
j
2
Spectrum at Jij=0
g

1
  
j
2
j

j
 , 
j
Two routes to quantization
B. Heisenberg spins


dS j
Action =  d  iSA S j
H
d
 j

 
First term is kinematic Berry phase which ensures
 S j , Sk    i jk   S j and S 2j  S ( S  1)
Spectrum at Jij=0
 ,
j
j
(2S+1)-fold degeneracy
Generalize model to SU(N) spins and
explore phase diagram in N, S plane
Outline
A. Insulating quantum rotors.
B. Insulating Heisenberg spins
C. DMFT of a random t-J model
D. Metallic spin glasses: DMFT of a
random Kondo lattice
A. Insulating quantum rotors
A. Quantum rotor model
 1

 dSi 
Action =  d   
   J ij Si  S j 
 2 g i  d  i  j

2
Jij : a Gaussian random variable with zero mean
T=0 phases
Local dynamic spin susceptibility
 n   
1/ T
0
        
d S j   S j  0 ein
            
Spin glass
Paramagnet
 ''   ~ qEA    
 ''   gapped
Specific heat C ~ T (?)
 ''   ~ 
g
D.A. Huse and J. Miller, Phys. Rev. Lett. 70, 3147 (1993).
J. Ye, S. Sachdev, and N. Read, Phys. Rev. Lett. 70, 4011 (1993).
N. Read, S. Sachdev, and J. Ye, Phys. Rev. B 52, 384 (1995).
A. Georges, O. Parcollet, and S. Sachdev, Phys. Rev. B 63, 134406 (2001).
T > 0 phase diagram



Quantum critical  ''   ~   

1/ 2
T
/
ln
1/
T
 

g
gc
J. Ye, S. Sachdev, and N. Read, Phys. Rev. Lett. 70, 4011 (1993).
N. Read, S. Sachdev, and J. Ye, Phys. Rev. B 52, 384 (1995).
B. Insulating Heisenberg spins
B. Heisenberg spin glass


dS j
Action =  d  iSA S j
  J ij Si  S j 
d i  j
 j

 
Jij : a Gaussian random variable with zero mean
S j : a SU(N ) spin with N 2  1 components and "length" S
S. Sachdev and J. Ye, Phys. Rev. Lett. 70, 3339 (1993).
T=0 phase diagram
S
Spin glass order
 ''   ~ qEA    
Specific heat C ~ T
(C ~ T2 ?)
Quantum critical "spin slush" phase
with "marginal Fermi liquid" spectrum:
sgn  
 ''   ~
J
N
S. Sachdev and J. Ye, Phys. Rev. Lett. 70, 3339 (1993).
A. Georges, O. Parcollet, and S. Sachdev, Phys. Rev. Lett. 85, 840 (2000).
A. Camjayi and M. J. Rozenberg, Phys. Rev. Lett. 90, 217202 (2003).
Quantum critical phase is described by
fractionalized S=1/2 neutral spinon excitations
S ~ f   f 
†
Spinon
spectral
density
 


  kBT 
1

S. Sachdev and J. Ye, Phys. Rev. Lett. 70, 3339 (1993).
T > 0 phase diagram
 

 kBT 
 ''   ~ sgn    
S. Sachdev and J. Ye, Phys. Rev. Lett. 70, 3339 (1993).
A. Georges, O. Parcollet, and S. Sachdev, Phys. Rev. Lett. 85, 840 (2000).
A. Camjayi and M. J. Rozenberg, Phys. Rev. Lett. 90, 217202 (2003).
C. Doping the quantum critical spin liquid
C. DMFT of a random t-J model
Hamiltonian =  tij Pc c j P   J ij Si  S j
†
i
ij
ij
1 †
Si  ci   ci
2
Jij : a Gaussian random variable with zero mean
O. Parcollet and A. Georges, Phys. Rev. B 59, 5341 (1999).
= carrier density
Quantum critical "incoherent" physics with universal  /k BT scaling above
a coherence scale
 F* ~ k BT * ~
 t 
2
J
O. Parcollet and A. Georges, Phys. Rev. B 59, 5341 (1999).
Physical consequences of quantum criticality
1. Electron spectral function (photoemission)
1  
Momentum integrated electron spectral density at T  0 :       * 
t  F 
1
1
    as   0 and    
as   

Momentum resolved
spectral density

2
Quasiparticle peak with residue Z ~ *

1

O. Parcollet and A. Georges, Phys. Rev. B 59, 5341 (1999).
Physical consequences of quantum criticality
2. d.c Resistivity
h T 
dc T   2   * 
e  F 
T

T 
 *
 F 
*
F
!
2
O. Parcollet and A. Georges, Phys. Rev. B 59, 5341 (1999).
Physical consequences of quantum criticality
3. NMR 1/T1 relaxation rate
1 1 T 
  * 
T1 J   F 
constant (MFL)
T 
Korringa  * 
 F 
O. Parcollet and A. Georges, Phys. Rev. B 59, 5341 (1999).
Physical consequences of quantum criticality
4. Optical conductivity
 F*   
In quantum critical regime, with   T  J , Re      

  k BT 
*
F
  F*
  for T    J
with Re      *
  F for   T
 T
O. Parcollet and A. Georges, Phys. Rev. B 59, 5341 (1999).
Phenomenological phase diagram for cuprates
O. Parcollet and A. Georges, Phys. Rev. B 59, 5341 (1999).
D. Metallic spin glasses
C. DMFT of a random Kondo lattice model
Hamiltonian =
t c
†
ij i
ij
c j   J ij Si  S j
JK

2
ij
 S c
i
†
i
  ci
i
Jij : a Gaussian random variable with zero mean
S. Sachdev, N. Read, and R. Oppermann, Phys. Rev. B 52, 10286 (1995).
A. M. Sengupta and A. Georges, Phys. Rev. B 52, 10295 (1995).
Quantum critical  ''   ~ sgn   
 ''   ~ qEA    sgn   
1/ 2
1/ 2
  
  3/ 2 
T 
JK
S. Sachdev, N. Read, and R. Oppermann, Phys. Rev. B 52, 10286 (1995).
A. M. Sengupta and A. Georges, Phys. Rev. B 52, 10295 (1995).
Outlook
• Spin glass order is an attractive candidate for a quantum critical point
in the cuprates, on both theoretical and experimental grounds.
(Impurities break the translational symmetry associated with chargeordered states, and the Imry-Ma argument then prohibits a quantum
critical point associated with charge order in the presence of
randomness in two dimensions)
• A simple mean-field theory of a doped Heisenberg spin glass naturally
reproduces all the “marginal” phenomenology.
• Needed: better theory of fluctuations in low dimensions